Question

a)In Chelmsford County Hospital, the probability that a randomly chosen birth results in a boy is 45%. Over the course of a month, there are 20 births at the hospital. Is a normal approximation to the binomial distribution appropriate here? Show your calculations and explain your choice.

b) In Chelmsford County Hospital, the probability that a randomly chosen birth results in a boy is 45%. Over the course of a month, there are 20 births at the hospital. What is the chance that more than 14 of the births are boys? Use the method of the normal approximation to the binomial distribution. Provide evidence of your work by typing out your full solution.

c)In Chelmsford County Hospital, the probability that a randomly chosen birth results in a boy is 45%. Over the course of a month, there are 20 births at the hospital. What is the chance that more than 14 of the births are boys? Use the binomial distribution formula. Provide evidence of your work by typing out your full solution.

Answer 1

a) If $X\sim Bin(n, p)$and if $n$ is large and/or $p$ is close to $1/2,$ then $X$ is approximately $N(np, npq),$ where $q=1-p.$

In practice, the approximation is adequate provided that both $np\geq10$ and $nq\geq 10,$ since there is then enough symmetry in the underlying binomial

distribution.

Sometimes consider the condition $np>5$ and $nq>5.$

If the binomial probability histogram is not too skewed, $X$ has approximately a normal distribution with $\mu=np$ and $\sigma=\sqrt{npq}$

$n=20, p=0.45$

$q=1-p=1-0.45=0.55$

Check the condition $np>5$ and $nq>5$

$np=20(0.45)=9>5$

$nq=20(0.55)=11>5$

We can safely approximate $\hat{p}$ by a normal distribution

$$\mu=np=20(0.45)=9$$

$$\sigma=\sqrt{npq}=\sqrt{20(0.45)(0.55)}=2.225$$

Check the condition $np\geq10$ and $nq\geq10$

$np=20(0.45)=9<10$

We cannot safely approximate $\hat{p}$ by a normal distribution.

(b) $n=20, p=0.45$

$q=1-p=1-0.45=0.55$

Check the condition $np>5$ and $nq>5$

$np=20(0.45)=9>5$

$nq=20(0.55)=11>5$

We can safely approximate $\hat{p}$ by a normal distribution

$$\mu=np=20(0.45)=9$$

$$\sigma=\sqrt{npq}=\sqrt{20(0.45)(0.55)}=2.225$$

$$P(X>14)=1-P(X\leq14)$$

$$=1-P(Z\leq\dfrac{14-\mu}{\sigma})=1-P(Z\leq\dfrac{14-9}{2.225})$$

$$\approx1-P(Z\leq2.247)\approx1-0.98768$$

$$\approx0.01232$$

c) $X\sim Bin(20, 0.45)$

$$P(X=x)=\dbinom{n}{x}p^x(1-p)^{n-x}$$

$$P(X>14)=P(X=15)+P(X=16)$$

$$+P(X=17)+P(X=18)+P(X=19)$$

$$+P(X=20)\approx0.00643$$